Ergodicity of a single number-field translation

Statement

For $\omega\in K_\infty^d$ , translation $T_\omega([x])=[x+\omega]$ on $X=(K_\infty/\mathcal O_K)^d$ with Haar probability is ergodic if and only if

\operatorname{Tr}_\infty(a\cdot\omega)\notin\mathbb Z \qquad\bigl(0\ne a\in(\mathcal O_K^\vee)^d\bigr).

Fourier proof

Inverse pullback satisfies

\alpha_\omega(\chi_a)=e^{-2\pi i\operatorname{Tr}_\infty(a\cdot\omega)}\chi_a.

The character is fixed exactly when its trace pairing is integral. Character completeness gives the fixed-function criterion.

Equivalent integral-coefficient test

The same ergodicity test can quantify over $0\ne k\in\mathcal O_K^d$ . Indeed, $\mathcal O_K\subseteq\mathcal O_K^\vee$ , and multiplying a violating codifferent frequency by an integer $N$ with $N\mathcal O_K^\vee\subseteq\mathcal O_K$ produces a violating integral frequency. This does not change which lattice indexes all Fourier characters.

Distinction from a ring action

The condition $k\cdot\omega\notin\mathcal O_K$ for nonzero integral $k$ is instead the criterion for the joint ring-of-integers translation action. It is insufficient for one translation in higher-degree fields.